Equilibria of the Curvature Functional and Manifolds of Nonlinear Interpolating Spline Curves

• Published in: SIAM journal on mathematical analysis Vol. 13; no. 3; pp. 421 - 458
• Main Authors: Golomb, Michael, Jerome, Joseph
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.05.1982
• Subjects: Equilibrium; Hypotheses
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/0513031

A detailed global and local analysis is carried out of smooth solutions of the variational problem \[ (1{\text{i}})\qquad \delta \int_0^s {\kappa ^2 (s)} ds = 0, \]subject to position function constraints \[ (1{\text{ii}})\qquad x(s_i ) = p_i ,\qquad 0 \leqq s_0 < s_1 < \cdots < s_m \leqq \bar s.\] Here $\{ p_i \} _0^m \subset \mathbb{R}^2 $ is prescribed, $x$ is a vector-valued function with curvature $\kappa (s)$ at arc length $s$ and the interpolation nodes $s_i $ are free. Problem (1) may be viewed as the mathematical formulation of the draftsman's technique of curve fitting by mechanical splines. Although most of the basic equations satisfied by these nonlinear spline curves have been known for a very long time, calculation via elliptic integral functions has been hampered by a lack of understanding concerning what precise information must be specified for the stable determination of a smooth, unique interpolant modeling the thin elastic beam. In this report, sharp characterizations are derived for the extremal interpolants as well as structure theorems in terms of inflection point modes which guarantee uniqueness and well-posedness. A certain type of stability is introduced and studied and shown to be related to (linearization) concepts associated with piecewise cubic spline functions, which have been studied for decades as a simplification of the nonlinear spline curves. Many examples are introduced and studied.